A114855 Expansion of q^(-1/3) * (eta(q) * eta(q^4))^2 / eta(q^2) in powers of q.

1, -2, 0, 0, 0, 4, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -17, 0, 0, 0, 0

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

S. Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 147, Ed. G. H. Hardy et al., AMS Chelsea 2000.

S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266. MR0099904 (20 #6340)

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

M. Josuat-Verges and J. S. Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, p. 28, equation (61), arXiv:1101.5608, 2011

Formula

Expansion of psi(x^2) * f(-x)^2 = phi(-x) * f(-x^4)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Euler transform of period 4 sequence [ -2, -1, -2, -3, ...].

a(n) = b(3*n + 1) where b(n) is multiplicative and a(p^e) = 0 if e is odd, a(3^e) = 0^e, a(p^e) = p^(e/2) if p == 1 (mod 3), a(p^e) = (-p)^(e/2) if p == 2 (mod 3).

Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = (u*w * (u + 2*w) * (u + 4*w))^2 - v^6 * (u^2 + 4*u*w + 8*w^2).

G.f.: Sum_{k} (3*k + 1) * x^(3*k^2 + 2*k) = Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)) * (1 - x^(4*k)).

a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0. a(4*n + 1) = -2 * a(n). 2 * a(n) = A113277(4*n + 1) = - A114855(4*n + 1).

(-1)^n * a(n) = A113277(n). a(8*n) = A080332(n).

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6^(3/2) (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - Michael Somos, Mar 11 2015

Example

G.f. = 1 - 2*x + 4*x^5 - 5*x^8 + 7*x^16 - 8*x^21 + 10*x^33 - 11*x^40 + ...
G.f. = q - 2*q^4 + 4*q^16 - 5*q^25 + 7*q^49 - 8*q^64 + 10*q^100 - 11*q^121 +...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^4]^2 EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Mar 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}];  (* Michael Somos, Mar 11 2015 *)
a[ n_] := With[{m = Sqrt[3 n + 1]}, If[ IntegerQ[ m], -m (-1)^Mod[ m, 3], 0]]; (* Michael Somos, Mar 11 2015 *)

Prog

(PARI) {a(n) = if( issquare( 3*n + 1, &n), n * -(-1)^(n%3), 0)};
(PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( e%2, 0, (-(-1)^(p%3) * p)^(e/2) )))) };
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A))^2 / eta(x^2 + A), n))};
(Magma) A := Basis( CuspForms( Gamma1(36), 3/2), 300); A[1] - 2*A[4]; /* Michael Somos, Mar 11 2015 */

Crossrefs

Cf. A080332, A113277, A114855.

Sequence in context: A246950 A204531 A113277 * A221381 A100951 A348616

Adjacent sequences: A114852 A114853 A114854 * A114856 A114857 A114858

Keyword

sign

Author

Michael Somos, Jan 01 2006

Status

approved